-sm|m| (1-)12+ (1+)12-Jn+ -() ,

and \(\sigma ^{\pm } = |m\pm \frac{1}{2}|\) , \(\xi = \mathrm {cos}\vartheta \) , \(s = \pm \) , \(n = 0,1,2, ...\) , \(m = \pm 1/2,3/2,5/2,... \)  and \(J_{n}^{\sigma ^{\pm }\sigma ^{\mp }}\)  are Jacobi polynomials [1]}. \(N_{n,m}\)  is a normalisation factor (given in Supplementary Material REF ). The corresponding discretized energy levels of these states are given by

Es, n, m  = s AR(n + |m|+ 12 ) ,

where \(A\)  is a material-dependent constant found from ab initio calculations [2]}. \(A\)  relates to the strength of spin-orbit coupling (SOC) in the material, averaged over the three Cartesian axes. This approximation of isotropic SOC allows for the analytical description of the topological surface states and their energy levels. Close to the \(\Gamma \) -point the energy levels are linearly spaced, and increase in degeneracy away from the Dirac point (i.e. \(2,4,6,8,... \) ) as illustrated in FIG. REF b. The energy values are inversely proportional to \(R\) , and for large \(R\)  we recover the expected continuum Dirac cone. Examples of the energy levels associated with quantum numbers \((s,n,m)\)  are given in FIG. REF c. The system of discrete energy levels can be tuned with particle size (much like the energy levels of a semiconductor quantum dot or topological insulator quantum dot), encouraging us to refer to this system as a topological quantum dot (TQD). The key differences between TQDs and semiconductor quantum dots are (i) the delocalization of states for TQDs vs the relatively localized states of the semiconductor counterpart, (ii) the quantisation of surface states rather than bulk states, which in turn affects (iii) the frequency-range of operation, which for semiconductor quantum dots is usually in the visible range and near infrared, but for TQDs is in the THz, e.g. for a Bi\(_2\) Se\(_3\)  TQD of \(5<R/\mathrm {nm}< 100\) , \(A=3.0\)  eVÅ [2]} and the spacing between two energy levels is \(0.72\mbox{-}14.4\)  THz, contrasting with a band gap of 220 meV \(\sim 53.2\)  THz.
